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www.cea.fr www.cea.fr
PDV: NOISE ANALYSIS
USING SHORT-TIME
FOURIER TRANSFORM
(STFT)
LIVERMORE, JUNE, 2016
Gabriel PRUDHOMME, CEA, DAM, DIF
PDV User Workshop
| PAGE 1
Gabriel PRUDHOMME | PDV and its noise | Livermore
WE CONSIDER THE FOLLOWING PDV DESIGN:
• Working @ 1.55 µm,
• Two laser configuration:
source and local oscillator.
The design is equivalent to a Michelson
interferometer.
(It is equivalent to have a reference mirror moving
with a constant velocity 𝑉𝐋𝐎 (equal to 𝜆𝑠/2 Δ𝑓))
𝑉𝐋𝐎
The target can be a solid
surface or a multiple
moving-object set.
Each object is
a reflector.
Introduction
3
beamspliter
Gabriel PRUDHOMME | PDV and its noise | Livermore
HIGH-VELOCITY PARTICLE CLOUDS IN SHOCK PHYSICS
• A metallic (tin) plate undergoes a shockwave (produced
by a laser shock, high-explosives or impact)
• Many mechanisms (spalling, microspalling, microjetting,
melting, …) generate a cloud of micron-sized particles
accelerating up to several km/s.
Introduction
4
Gabriel PRUDHOMME | PDV and its noise | Livermore
THE NEED: PARTICLE CLOUDS
Particle cloud
(Rear face) free
surface velocity
Are the fastest particles detected?
How deep is the cloud analyzed?
Noise on spectrogram may limit the visibility of the cloud by PDV.
Noise is linked to the PDV design.
𝜑(𝑡, 𝑣)
Tin micro-jetting first PDV measurement at the CEA
(Mercier et al., 2006)
Ve
locity [
m/s
]
Time [µs]
Spectrogram noise
Mean level: 𝐵𝜑
Introduction
5
Gabriel PRUDHOMME | PDV and its noise | Livermore
DSP
SHORT-TIME FOURIER TRANSFORM
FT
In this talk, we deal with STFT
(Short-Time Fourier Transform),
which leads to strong
consequences.
Introduction
6
Gabriel PRUDHOMME | PDV and its noise | Livermore
HOW DO WE SEPARATE THE NOISE SOURCES?
(Ph): shot noise • Quantum optics : the photons
are inseparable amounts of
power.
(D): Dark noise Includes background IR (BLIR)
noise, electronic noise, thermal
noise, …
(Q): digitizer noise (S): spontaneous-emitted
photons – ASE noise
• Due to the lasers and
amplifiers of the PDV line.
• Generally expressed as a
number of “equivalent bit”.
• We will express all noise sources as an optical “NEP” (Noise Equivalent Power”,
unit : 𝑊 𝐻𝑧 ).
• This choice will be explained in the 2nd part.
• Relation between optical power 𝑷𝒅𝒆𝒕 and electrical 𝒖𝒆 : 𝒖𝒆 = ℛ 𝑷𝒅𝒆𝒕.
• First, we derive the power 𝑷𝒅𝒆𝒕 on the PDV detector in order to estimate noise levels.
What are the noise sources of the standard PDV?
8
Gabriel PRUDHOMME | PDV and its noise | Livermore
BACK TO BASICS: THE OPTICAL POWER
COLLECTED BY THE DETECTOR
• For a single free surface: 𝑃𝑑𝑒𝑡(𝑡) = 𝛼𝐿𝑂𝑃𝐿𝑂 + 𝛼𝐿𝑂𝑃𝐿𝑂𝛼𝑜𝑃𝑝 𝜅𝐿𝑂𝑝cos Ω𝑝𝑡 + 𝜙𝐿𝑂
𝑝+ 𝑎𝑜𝑃𝑝
• F𝐨𝐫 𝑵 objects (i.e, a particle cloud), in order of importance:
𝑃𝑑𝑒𝑡(𝑡) = 𝛼𝐿𝑂𝑃𝐿𝑂(1)
+ 𝛼𝐿𝑂𝑃𝐿𝑂𝛼𝑜𝑃𝑝 𝜅𝐿𝑂𝑝cos Ω𝑝𝑡 + 𝜙𝐿𝑂
𝑝
𝑝
(2)
+ 𝛼𝑜 𝑃𝑝𝑃𝑞 𝜅𝑝𝑞cos Ω𝑝 − Ω𝑞 𝑡 + 𝜙𝑝
𝑞
𝑝,𝑞
(3)
𝑝 (or 𝑞) : points to a reflector (i.e. a free surface or a particle),
𝛼𝐿𝑂, 𝛼𝑜: coupling efficiencies,
𝜅𝑝𝑞: polarization dot product (between 𝑝 and 𝑞),
Ω𝑝 =4𝜋
𝜆𝑃𝐷𝑉𝑣𝑝 − 𝑉𝐿𝑂 : relative angular frequency.
(1) constant term, strong but useless.
≈ few 100 µw? (2) beating term (LO coupling), weak but that’s we are looking for!
≈ (few 100 µw)(? to 1 µW?) ≲ 10 µ𝑊 for each 𝑝.
𝑁 × few 10 µ𝑊 spread over the velocity spectrum.
(3) cross-beating term, very weak, neglected in general!
𝑁2 × (≲ 1 µ𝑊) : weak when 𝑁 is not too large.
the sum could be separated in two parts:
- (3c) 𝑝 = 𝑞, 𝛼0 𝑃𝑝𝑝 : constant part.
- (3n) 𝑝 ≠ 𝑞, 𝛼0 𝑋𝑝𝑞cos Ω𝑝 − Ω𝑞 𝑡 + 𝜙𝑝
𝑞 (𝑝≠𝑞) : (nearly) random freq. -> noise ?!
9
What are the noise sources of the standard PDV?
Gabriel PRUDHOMME | PDV and its noise | Livermore
WHICH TERMS ARE IMPORTANT FOR THE NOISE? 1/4
Shot noise (or photonic noise)
Because a lot of beats is recorded inside the
FT window 𝑇 ≫2𝜋
Ω𝑝, we consider the
constant part of the signal:
𝑃𝑑𝑒𝑡 = 𝛼𝐿𝑂𝑃𝐿𝑂(1)
+ 𝛼𝑜 𝑃𝑝𝑝
(3𝑐)
If 𝜶𝒐 𝑃𝑝𝑝 ≤ 10 µW, we suppose that 𝑷𝒅𝒆𝒕 is
constant and equal to (1).
The shot-noise spectrum density is:
< 𝑃𝑝ℎ > = 𝐾𝑝ℎ 𝑓𝐵𝑊 =2 ℎ𝑐0 𝑓𝐵𝑊 𝛼𝐿𝑂𝑃𝐿𝑂𝜂 𝜆𝑃𝐷𝑉
𝐾𝑝ℎ =2 ℎ𝑐0 𝛼𝐿𝑂𝑃𝐿𝑂𝜂 𝜆𝑃𝐷𝑉
200
250
300
350
400
450
500
550
600
0,1 10 1000
To
tal m
ean
po
wer ⟨𝑃
𝑑𝑒𝑡
⟩ [
µW
]
Total object power ΣPp [µW]
Total intensity on detector (CEA PDV System)
10
What are the noise sources of the standard PDV?
Gabriel PRUDHOMME | PDV and its noise | Livermore
WHICH TERMS ARE IMPORTANT FOR THE NOISE? 2/4
Dark noise
Values taken from datasheet (NEP),
Or measured.
< 𝑷𝑫 > = 𝑲𝑫 𝒇𝑩𝑾 𝒇𝑩𝑾
𝑲𝑫 𝒇𝑩𝑾 is the NEP of the detector.
It depends on the detector technology.
In practice, it depends on the detector
bandwidth.
1 10 100
KD (
~N
EP
)
Velocity dynamics ΔV (= v-VLO) [km/s]
NEP from a product line of detectors
Best
11
What are the noise sources of the standard PDV?
Gabriel PRUDHOMME | PDV and its noise | Livermore
WHICH TERMS ARE IMPORTANT FOR THE NOISE? 3/4
Digitizer noise
The constant term (1) may be canceled
with an offset
The beating term (2) will set the dynamics
of the digitizer, and its noise level.
Mainly used characteristic to estimate
digitizer noise : ENOB (effective number
of bits), composed of:
noise floor level,
dynamic noise level,
distortions (total harmonic – TH).
𝐾𝑄 =Δ𝑈𝑃−𝑃
2 3 2𝐸𝑁𝑂𝐵 − 1 𝑓𝐵𝑊
NB : distortions create harmonics, it is
not so disadvantageous for free surface
velocities
12
What are the noise sources of the standard PDV?
Gabriel PRUDHOMME | PDV and its noise | Livermore
WHICH TERMS ARE IMPORTANT FOR THE NOISE? 4/4
ASE noise
Generated by laser source, local
oscillator and optical amplifier (EDFA).
We do not use EDFA.
We use very high-coherency laser
(frequency width < 100 kHz), the
spontaneous emission is really weak!
For our PDV systems,
𝑲𝑺 = 𝟎
13
What are the noise sources of the standard PDV?
Gabriel PRUDHOMME | PDV and its noise | Livermore
TOTAL « NEP » 𝑲𝑻 OF THE PDV SYSTEM IS:
𝐾𝑇2 = 𝐾𝑄
2 + 𝐾𝐷2 + 𝐾𝑝ℎ
2 + 𝐾𝑆2
You can’t act on
it ! Changing 𝑃𝐿𝑂? → Backup
The detector
may be changed
The digitizer
may be changed
0,1 1 10
Kx
Maximum velocity dynamics ΔV (= v-VLO) [km/s]
How to select a couple of {detector + digitizer}?
Digitizer Detector
BE
ST
14
What are the noise sources of the standard PDV?
Gabriel PRUDHOMME | PDV and its noise | Livermore
NOISE SOURCE LEVELS (8-GHZ CEA PDV SYSTEM)
Example with a 8-GHz bandwidth, CEA PDV system
(𝛼𝐿𝑂𝑃𝐿𝑂 = 250 µW).
1. Photonic noise – 1.1 µW For low reflected power (𝑃𝑜 < 10 µ𝑊), it depends only on 𝛼𝐿𝑂𝑃𝐿𝑂 and detector efficiency.
2. Internal detector noise – 2.1 µW Depends on the detector.
3. Quantification noise – 2.3 µW Depends on the total reflected power and on the digitizer.
4. AES noise – 0 µW Very spectral-narrow lasers and no optical amplifier.
So, 𝐵𝑑𝑒𝑡 = 1.12 + 2.12 + 2.32 = 3.3 µ𝑊
What are the noise sources of the standard PDV?
Curves of noise sources as a function of the vertical resolution (backup)
15
Gabriel PRUDHOMME | PDV and its noise | Livermore
WHY HETERODYNING IS SO SENSITIVE ?
Let’s define a “NEP” in object space : “object-equivalent noise”
…(see my thesis)
𝐵𝑜𝑒 =𝐾𝑇2 𝑓𝐵𝑊
8𝛼𝑜𝛼𝐿𝑂𝑃𝐿𝑂= 5.3 nW (𝛼𝐿𝑂𝑃𝐿𝑂 = 250 µW)
Another reinterpretation of the high detectivity of heterodyne systems:
𝐵𝑜𝑒 ≪ 𝐵𝑑𝑒𝑡, thanks to the P𝐿𝑂𝑃𝑃 term.
“Only objects, of which reflected and collected power is above
𝐵𝑜𝑒, could be measured (using the raw signal)”
𝐵𝑒𝑜 = 5.3 𝑛𝑊 𝐵𝑑𝑒𝑡 = 𝐾𝑇 𝑓𝐵𝑊= 3.3 µW
Target PDV
Recorder
Probe
𝛼𝐿𝑂𝑃𝐿𝑂= 250 µW
17
Gabriel PRUDHOMME | PDV and its noise | Livermore
WHY {HETERODYNING+STFT} ARE REALLY SO SENSITIVE?
• Fourier transform consequences:
– Mono frequency base: 𝑒4𝑖𝜋 𝑣𝑚𝑡𝜆𝑠 -> monokinetic
contributors are identified for each velocity (𝑣𝑚).
– Noise is spread on all the bandwidth: after “𝐹𝐹𝑇2”, noise mean level is reduced by 𝑁.
“≈ (𝑆𝐼𝐺𝑁𝐴𝐿)2+ (𝑁𝑂𝐼𝑆𝐸)2
𝑁 ”
The signal is spectrally localized.
(𝑁𝑂𝐼𝑆𝐸)2
𝑁
The noise is spread on
the bandwidth
SIGNAL+NOISE
For example, see Harris, 1978, Proc IEEE
The Effects of STFT transform on noise
Gabriel PRUDHOMME | PDV and its noise | Livermore
NOISE ON THE
SPECTROGRAM: 𝑩𝝋
Taking into account FT, noise
level on the spectrogram is
𝐵𝜑 =1
𝐺 𝑖 𝑇𝑓BW 𝐵𝑒𝑜
=1
𝐺 𝑖 𝑇
1
8𝛼𝑜𝛼𝐿𝑂𝑃𝐿𝑂𝐾𝑄2 + 𝐾𝐷
2 + 𝐾𝑝ℎ2
noise levels
𝐺 𝑐 is a window correction factor (≈ 0.2 ).
STFT makes detection limit independent of
bandwidth (if Nyquist criteria is respected).
- but 𝐾𝑋 are hardware-depending.
- It is useless to pre-filter the signal with a low-
pass filter (maybe to reduce the number of
samples).
The Effects of STFT transform on noise
Gabriel PRUDHOMME | PDV and its noise | Livermore
NOISE ON SPECTROGRAMS: COMPARISONS
𝐵𝜑 =1
𝐺 𝑖 𝑇𝑓BW 𝐵𝑒𝑜 =
1
𝐺 𝑖 𝑇
1
8𝛼𝑟𝛼𝑜𝑃𝑟𝐾𝑄2 + 𝐾𝐷
2 + 𝐾𝑝ℎ2
noise levels
– When no signal is present, the noise PDF* is
exponential.
– 𝐵𝜑 is both the bias and the dispersion of the
background noise (The mean is equal to the
standard deviation).
• *PDF: Probability Density Function
Analyzed
area
For 3 windows of 𝑇 width, histograms
computed from spectrograms before
shock break-out are analyzed.
(10% max. error)
The Effects of STFT transform on noise
exp PDF
Gabriel PRUDHOMME | PDV and its noise | Livermore
HOW TO IMPROVE SPECTROGRAM DISPLAYS?
exp PDF
z → 1 − exp −𝑧
𝐵𝜑 CDF : relative
number of background values below 𝑧.
So, s → exp −𝑠
𝐵𝜑 gives the relative number of
« noisy » visible points.
Example : 𝑇 = 50 𝑛𝑠 @ 50 GS/s,
𝑁 = 2500
𝑆𝜑 = 5𝐵𝜑 → ≈ 0.67 %
17 expected noisy points in each
column. 0,01%
0,10%
1,00%
10,00%
1Bφ 3Bφ 5Bφ 7Bφ
rela
tive n
um
ber
of
« n
ois
y »
vis
ible
po
ints
s : threshold
Which threshold should be set to display spectrogram?
The Effects of STFT transform on noise
How many “blue points” do we accept?
22
Gabriel PRUDHOMME | PDV and its noise | Livermore
HOW TO IMPROVE SPECTROGRAM DISPLAYS?
EXAMPLES 𝑆𝜑
𝐵𝜑 𝐞𝐱𝐩 −
𝑆𝜑
𝐵𝜑
1 36 %
3 5,0 %
5 0,67 %
7 0,091 % 𝑆𝜑
𝐵𝜑= 1
𝑆𝜑
𝐵𝜑= 3
𝑆𝜑
𝐵𝜑= 5
The Effects of STFT transform on noise
23
Gabriel PRUDHOMME | PDV and its noise | Livermore
SUMMARY
We present rules for PDV design in order to decrease
noise:
Contributors are quantified.
Major ones: detector background and digitizer noise.
Other source noises? (non-linear effect, backscattering inside the fiber, …)
Our model provides a good estimation of 𝐵𝜑, the spectrogram mean noise level.
We analyzed the impact of noise on STFT spectrograms
Noise level depends on noise density (“NEP”) and not on bandwidth,
Reducing digitizer bandwidth will not decrease 𝐵𝜑 (as long as Nyquist is
respected).
We are able to threshold the spectrogram as a function of the number of noisy
samples.
24
Commissariat à l’énergie atomique et aux énergies alternatives
CEA, DAM, DIF | 91297 ARPAJON
Etablissement public à caractère industriel et commercial | RCS Paris B 775
685 019
THANK YOU FOR YOUR ATTENTION
| PAGE 25
Gabriel PRUDHOMME | PDV and its noise | Livermore
𝑃𝑳𝑶 MAY BE INCREASED?
• Why? 𝑃𝐿𝑂 ↗, gain ↗ ( 𝑃𝐿𝑂), detectivity ↗ because CEA PDV system is not photonic-noise limited.
• Detector limitation: – 1 mW for saturation.
– ≳ 500 µW for non-linear effects, but may be corrected by post-processing (required 𝑃𝐿𝑂 or 𝑃𝑑𝑒𝑐𝑡 knowledge).
• 𝑃𝑑𝑒𝑡,max ≈ 𝛼𝐿𝑂 ≈2%
𝑃𝐿𝑂 + 𝛼𝐿𝑂 𝛼𝑜 ≈1
𝑃𝐿𝑂𝑃𝑜, 𝑃𝑜 > 1 µ𝑊 due to circulator leaks.
Technical development required.
• Up to 4x 𝑃𝑟 increasing is relevant (𝐾𝑝ℎ = 𝐾𝑞).
– 2x: linear domain.
– 4x: non-linear domain.
27
[
Gabriel PRUDHOMME | PDV and its noise | Livermore
NOISE AS A FUNCTION OF DIGITIZER V. RESOLUTION (CEA PDV
SYSTEM)
28
Gabriel PRUDHOMME | PDV and its noise | Livermore
“PDV RADIOMETRIC SCHEME”
Radiometric PDV equation of the spectrogram:
𝜑 𝑡, 𝑣 =1
2 ℛ𝑠𝑦𝑠2 𝑃𝐿𝑂
1
𝐺𝑖 𝑁2STFT 𝑡𝑓=2𝑣 𝜆
𝑢2
ℛ𝑠𝑦𝑠2 : system response,
𝑃𝐿𝑂 : local oscillator power,
𝐺𝑖 𝑁2 : FT normalization, ≈ bin number of the window (𝑁).
Equals to the power distribution over the
velocities.
Leads to the definition of a spectrogram mean
noise level 𝐵𝜑.
Do not take into account the polarization-
efficiency (𝑒 𝑜 ⋅ 𝑒 𝐿𝑂) dot factor. Let’s include it
into ℛ𝑠𝑦𝑠. (see J-E. FRANZKOWIAK’s talk).
The point is:
The dB-expressed DSP is equal to the
optical power distribution (with an offset).
29